Modern Algebra (M.S. and Ed.D.)

Preparatory Courses: Math 4613, 5013

1.

Groups: Equivalence relations, cyclic groups, subgroups, quotient groups, homomorphisms, direct products, symmetric groups, normal subgroups, finite groups, Sylow theorems, fundamental theorem of finitely generated abelian groups. Examples include the integers, the reals, the rationals, the complex numbers, quaternions, $A$_n, S_n, D_n, finite abelian groups, cyclic groups, all groups of order less than or equal to 12, and matrix groups

2.

Vector spaces: Examples, linear transformations, independence and dependence, bases, matrices, canonical forms, eigenvalues, eigenvectors

3.

Rings and modules: Homomorphisms, integral domains, division rings, field of quotients, PIDs, polynomial rings, modules over PIDs

4.

Fields: Extension fields, splitting fields, Galois theory in characteristic 0, solvability by radicals

REFERENCES: Complete coverage is found in: E.A. Walker, Introduction to Abstract Algebra, Chapter 1, Chapter 2, Chapter 3 (Sections 1-3), Chapter 4, Chapter 5 (Sections 1-3), Chapter 6, Chapter 7 (Section 3), or I. N. Herstein, Topics in Algebra, Chapters 1-3, Chapter 4 (Sections 1,2,5), Chapter 5 (Sections 1,3,6,7), Chapter 6 (Sections 1-7). The material on groups is found in: D. Saracino, Abstract Algebra, Chapters 0-15.